Rocks have a range of pore sizes. Hydrocarbons have a range of components. The pore-size distribution and component distribution give rise to distributions of relaxation times and diffusion coefficients observed with nuclear magnetic resonance (NMR). Estimation of these distributions from the measured magnetisation data plays a central role in the inference of in situ petro-physical and fluid properties. From the estimated distributions, linear functionals are computed to provide insight into rock or hydrocarbon properties. For example, moments of relaxation time or diffusion are often related to rock permeability and/or hydrocarbon viscosity. Specific areas underneath the relaxation time or diffusion distribution are used to estimate fluid saturations as well as bound and free fluid volumes.
In oilfield applications, the measured NMR magnetization data denoted by G(t) is a multi-exponential decay, with time constant T2 and amplitudes ƒ(T2).
                              G          ⁡                      (            t            )                          =                              ∫            0            ∞                    ⁢                                                    P                τ                            ⁡                              (                                  T                  2                                )                                      ⁢                          e                                                -                  t                                /                                  T                  2                                                      ⁢                          f              ⁡                              (                                  T                  2                                )                                      ⁢                                          dT                2                            .                                                          (        1        )            
The relaxation time T2 is the characteristic time corresponding to loss of energy by protons in hydrocarbons or water present in pores of a rock or in the bulk fluid. The function Pτ(T2) is referred to as polarization factor and depends on the pulse sequence. For example,
            P      τ        ⁡          (              T        2            )        =      {                            1                                      C            ⁢                                                  ⁢            P            ⁢                                                  ⁢            M            ⁢                                                  ⁢            G            ⁢                                                  ⁢            pulse            ⁢                                                  ⁢            sequence            ⁢                                                  ⁢            with            ⁢                                                  ⁢            full            ⁢                                                  ⁢            polarization                                                            1            -                          2              ⁢                              e                                                      -                    τ                                    /                                      T                    2                                                                                                                          inversion              ⁢                                                          ⁢              recovery                        -                          C              ⁢                                                          ⁢              P              ⁢                                                          ⁢              M              ⁢                                                          ⁢              G              ⁢                                                          ⁢              pulse              ⁢                                                          ⁢              sequence                                                                        1            -                          e                                                -                  τ                                /                                  T                  2                                                                                                        saturation              ⁢                                                          ⁢              recovery                        -                          C              ⁢                                                          ⁢              P              ⁢                                                          ⁢              M              ⁢                                                          ⁢              G              ⁢                                                          ⁢              pulse              ⁢                                                          ⁢              sequence                                          where τ is a function of pre-polarization time Tw and longitudinal relaxation T1. In downhole applications, the function Pτ(T2) is a complex function of tool geometry (such as length of magnet and design of RF antenna), operational constraints (such as cable speed) as well as the pulse sequence.
In all applications, it is assumed that the data G(t) are measured with additive, white, Gaussian noise. The measurement domain is the time domain. Answer products are calculated from the measured G(t) as follows. Traditionally, assuming Pτ(T2) is known, an inversion algorithm is used to estimate the distribution of relaxation times ƒ(T2) in eqn. (1) from indications of the measured data G(t). Next, linear functionals of the estimated ƒ(T2) are used to estimate the petro-physical or fluid properties. For example, the area under the T2 distribution is interpreted as the porosity of the rock. Often, based on lithology, a threshold T2 is chosen as the cut-off characteristic time separating fluid bound to the pore surface and fluid that is not bound to the pore surface and can flow more easily. For example, in sandstones, the area under the T2 distribution corresponding to relaxation times smaller than 33 msec has been empirically related to bound fluid volume. The remaining area under the T2 distribution corresponds to free fluid volume. The mean of the distribution, ln(T2), is empirically related to either rock permeability and/or to hydrocarbon viscosity. The width of the distribution, σln(T2), provides a qualitative range of the distribution of pore sizes in the rock. Moments of relaxation time or diffusion are often related to rock permeability and/or hydrocarbon viscosity. Similar answer products can also be obtained from linear functionals computed from two-dimensional diffusion-relaxation data or T1-T2; relaxation data.
Estimation of ƒ(T2) is an ill-conditioned and non-linear problem. Small changes in the measured data G(t) due to noise can result in widely different ƒ(T2). In theory, there are infinitely many ƒ(T2) that fit the data. In the literature, this problem is addressed by choosing the “smoothest” solution ƒ(T2) that fits the data. This smooth distribution is often estimated by minimization of a cost function Q with respect to the underlying ƒ,Q=PG−LƒP2+αPƒP2,  (2)where G is the measured data, L is the matrix of the discretized function Pτ(T2)e−t/T2 and ƒ is the discretized version of the underlying density function ƒ(T2) in eqn. (1). The first term in the cost function is the least squares error between the data and the fit from the model in eqn. (1). The second term is referred to as regularization and incorporates smoothness in the relaxation amplitudes into the problem formulation. The parameter α denotes the compromise between the fit to the data and an a priori expectation of the distribution. The use of regularization and incorporation of smoothness into the problem formulation is subjective. In eqn. (2), it is possible to define alternate cost functions based on entropy, the Backus-Gilbert method or other measures of difference between data and fit.